Joyal's CatLab
Functors

**Basic Category Theory** ## Contents * Categories * Functors * Natural transformations * Functors of two variables * Adjoint Functors and Monads

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**Category theory** ##Contents * Contributors * References * Introduction * Basic category theory * Weak factorisation systems * Factorisation systems * Distributors and barrels * Model structures on Cat * Homotopy factorisation systems in Cat * Accessible categories * Locally presentable categories * Algebraic theories and varieties of algebras

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Contents

Covariant functors

The composite of two functors F:CDF:\mathbf{C}\to \mathbf{D} and G:DEG:\mathbf{D}\to \mathbf{E} is a functor GF:CEG F:\mathbf{C}\to \mathbf{E}. There is also an [identity functor] I C:CCI_{\mathbf{C}}:\mathbf{C}\to \mathbf{C} for every category C\mathbf{C}. Hence the categories form a category. A functor F:CCF:\mathbf{C}\to \mathbf{C} is an [isomorphism] if it is invertible in this category.

Examples

H *(;A):TopAb *H_*(- ;A): \mathbf{Top} \to \mathbf{Ab}_*

by putting H *(f;A)=f *H_*(f ;A)=f_* for every continuous map ff.

Special functors

Functors with special properties are important in applications. A functor F:CDF:\mathbf{C}\to \mathbf{D} is said to be

A functor F:CCF:\mathbf{C}\to \mathbf{C} is an isomorphism iff it is fully faithful and the map Ob(C)Ob(D)Ob(\mathbf{C})\to Ob(\mathbf{D}) induced by FF is bijective.

Perhaps less ubiquitous are:

When an inclusion is full, its domain is called a [full subcategory] of its codomain.

Contravariant functors

The composite of two functors F:CDF:\mathbf{C}\to \mathbf{D} and G:DEG:\mathbf{D}\to \mathbf{E} is a functor GF:CEG F:\mathbf{C}\to \mathbf{E} which is

A covariant functor F:CDF:\mathbf{C}\to \mathbf{D} induces a covariant functor between the opposite categories F o:C oD oF^o:\mathbf{C}^o\to \mathbf{D}^o. By definition, the functor F oF^o takes an object A oC oA^o\in \mathbf{C}^o to the object (FA) oD o(F A)^o\in \mathbf{D}^o and takes a morphism f o:A oB of^o:A^o\to B^o to the morphism F(f) o:(FA) o(FB) oF(f)^o: (F A)^o\to (F B)^o. We shall say that the functor F oF^o is the opposite of the functor FF. Similarly, a contravariant functor F:CDF:\mathbf{C}\to \mathbf{D} induces a contravariant functor between the opposite categories F o:C oD oF^o:\mathbf{C}^o\to \mathbf{D}^o.

For any category C\mathbf{C}, the map CC o\mathbf{C}\to \mathbf{C}^o which takes an object ACA\in \mathbf{C} to the opposite object A oC oA^o\in \mathbf{C}^o and takes a morphism f:ABf:A\to B to the opposite morphism f o:B oA of^o:B^o\to A^o is a contravariant functor Op:CC oOp:\mathbf{C}\to \mathbf{C}^o. We shall use the same notation for the inverse functor Op:C oCOp:\mathbf{C}^o\to \mathbf{C}. If F:CDF:\mathbf{C}\to \mathbf{D} is a contravariant functor, then the composite FOp:C oDF Op:\mathbf{C}^o\to \mathbf{D} is a covariant functor. The map FFOpF\mapsto F Op induces a bijection between the contravariant functors CD\mathbf{C}\to \mathbf{D} and the covariant functors C oD\mathbf{C}^o\to \mathbf{D}. Similarly, the map FOpFF\mapsto Op F induces a bijection between the contravariant functors CD\mathbf{C}\to \mathbf{D} and the covariant functors CD o\mathbf{C}\to \mathbf{D}^o.

Examples

Revised on February 18, 2011 at 10:23:16 by Tom Hirschowitz